Yesterday I was in the mall and there is an Apple store in there so....I took my iphone 5 ios 6.1.3 in to the techs. Here is what I was told and how it was fixed: My update had gotten corrupted and the modem was bombing out. So that is what was draining my phone. I hadn't used it and it was at 80% at 10am. The fix was to backup to icloud (which I had done) and then completely restore the phone to factory settings. The tech did it through itunes, so the phone was now plugged in. He said my update was corrupted by doing it "over the air" and by not having it plugged in to a power source. After the restore and all my apps reloaded back onto the phone I was good to go.
However, I noticed that my battery was still draining quickly last night, so I let it get down pretty low and then turned off all icloud settings and did a hard reset (hold down home button and on/off button, until apple appears).Then I turned back on all the icloud settings. (I really don't know if this makes any difference) Then I plugged it in to charge for the night.
It is now 11:27am and I am still at 96% and have used the phone for texting and email. So I have my fingers crossed that my battery issue is fixed. If the problem is all these phones got corrupted updates, then Apple isn't going to be putting out an another update so fast.
So I know this is another fix and it takes some time, but it worked for me, for now. I will update if my phone seems to resort back to draining again. Good luck!